Sensitivity and specificity enhancement in medical imagingplc/ammari2012.pdfphoto-acoustic imaging...

Sensitivity and specificity enhancement inmedical imaging

Habib Ammari

Department of Mathematics and ApplicationsEcole Normale Superieure, Paris

Sensitivity and specificity enhancement in medical imaging Habib Ammari

Sensitivity and specificity in medical imaging

• Mathematical and numerical modelling in medical imaging of cancer.

• Early detect tumors and determine which are malignant and which arebenign.

• Wave imaging of cancer tumors: elastic, optical, electric contrasts;specific dependence with respect to the frequency.

• Contrasts depend on molecular building blocks and on the microscopicand macroscopic structural organization of these blocks.

• Enhance the specificity and sensitivity of cancer detection.

Sensitivity and specificity• Single wave imaging: sensitivity to only one contrast.

• Spatial resolution: determined by the wave propagation phenomena andthe sensor technology.

• Multi-wave imaging: one single imaging system based on the combineduse of two kinds of waves.

• One wave will give its contrast and the second its spatial resolution.

• Wave 1 (high contrast + low resolution) + Wave 2 (low contrast + highresolution) = Image (high contrast + high resolution).

Image of breast cancer.

Multi-wave medical imaging

• 3 kinds of interactions between waves:

• Interaction of Wave 1 with tissues generates Wave 2:photo-acoustic imaging (V. Jugnon), thermo-acoustic imaging;

• Wave 1 can be tagged locally by Wave 2: acousto-opticaltomography, Electrical impedance tomography with ultrasound(with E. Bonnetier, Y. Capdeboscq, M. Tanter, and M. Fink);

• Wave 1 (travelling much faster than Wave 2) can be used toproduce a movie of Wave 2: elastography (P. Garapon).

• Imaging systems developed at Institut Langevin.

One single wave imaging

• One single wave imaging: bio-inspired approaches.

• Super-resolution in electro-sensing (with T. Boulier, J. Garnier, W. Jing,H. Kang, and H. Wang).

• Weakly electric fishes possess: one electro-emitter and manyelectro-receptors of 2 types. One type measures the amplitude of theelectric field and another measures its phase.

Blackghost Knifefish (weakly electric: 1mV ).

• Source term f time periodic and separable: f (x , t) = f (x)∑

n einω0t ; ω0:

fundamental frequency.

• Target D = z + δB; z : location; δ: characteristic size of the target;k = (σ + iωε)/σ0; k, σ, and ε: the admittivity, the conductivity, and thepermittivity of the target; ω = nω0: the probing frequency.

• u : the electric potential field generated by the fish:

∆u = f , x ∈ Ω,

∇ · (1 + (k − 1)χ(D))∇u = 0, x ∈ Rd \ Ω,

∣∣∣∣−

= 0, x ∈ ∂Ω,

[u] = ξ∂u

∣∣∣∣+

, x ∈ ∂Ω,

|u(x)| = O(|x |−d+1), |x | → ∞.

• ξ: effective thickness.

• The effective thickness ξ = δσ0/σs .

• σ0 ∼ 0.01S ·m−1; σb = 1S ·m−1 (highly conductive);

• Skin: very thin (δ ∼ 100µm) and highly resistive (σs ∼ 10−4S ·m−1).

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In the absence of any target In the presence of a target(ξ = 0.1). (δ = 0.2, σ = 2, ε = 0).

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The 8 elements of the dictionary.The dotted lines indicate a target with different electrical parameters.

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The real part of the electric field is plotted,for 4 (over 20) positions that the fish takes around the target (placed at the origin).

• Dipole approximation:

• u(x)− U(x) ' p · ∇G (x − z).• G : Green’s function.• p: dipole moment

p = −M(k ,D)︸︷︷︸∇U(z)

Polarization Tensor

• Source term f real: =mu(x) ' (=mp) · ∇G (x − z).

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• Polarization tensor:

∫∂D

x(λI −K∗D)−1[ν](x) dσ(x),

• λ = (k + 1)/(2(k − 1)): k: conductivity contrast;

• K∗D (Neumann-Poincare operator): weakly singular integral operator. K∗Dcompact (in the smooth case): discrete spectrum in (−1/2, 1/2) with 0as an accumulation point.

• 0 < k 6= 1 <∞: λI −K∗D : L2(∂D)→ L2(∂D) invertible.

Near-field imaging

• Polarization tensor: low frequency information; mixture of materialparameter and size.

• Multipolar approach (at a single frequency):

• Use the dipole + quadrupole approximation of the target.• Construct shape descriptors invariant with respect to

translation, rotation, and scaling (in two and threedimensions).

• Multipolar approximation:

u(x)− U(x) '∑α,β

(∂αG(x − z)Mαβ(k,D)∂βU(z).

• Mαβ(k,D): high-order polarization tensors.

• Reconstruction of high-order polarization tensors from the data by a leastsquares method.

• Instability:

Mαβ(k,D) = O(|D||α|+|β|+d−2), |∂αG(x−z)| = O(|x |−|α|−d+2)(|x | → +∞)

• Resolving power= number of high-order polarization tensors reconstructedfrom the data: depends on the signal-to-noise ratio (SNR) in the data.

• ε = characteristic size of the target/ the distance to the array oftransmitters/receivers.

• SNR = ε2/standard deviation of the measurement noise (Gaussian).

• Formula for the resolving power m as function of the SNR:

(mε1−m)2 = SNR.

Near-field imaging

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Classification from multipolar measurements with 10% (measurement) noise.

Multi-frequency imaging

• Multi-frequency approach: ω 7→ M(k(ω),D).

• Invariance with respect to translation, rotation, and scaling.• λj(ω): singular values of M(k(ω),D); ω∞: highest probing

frequency. Plot

ω 7→ λj(ω)

λj(ω∞),

for j = 1, . . . , d .

Multi-frequency imaging

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Classification from multi-frequency measurements with 10% noise.

Nano-particles for imaging

• Spectral decomposition (D smooth): K∗D [ψn] = µnφn.

• Plasmonic resonances: µn.

• ψn, φn: plasmonic eigenvectors.

• Gold nano-particles (negative conductivity):

k(ω) =2µn + 1

2µn − 1+ iτ ;

• τ : Debye relaxation term (small).

• Blow-up of the polarization tensor:

∫∂D

x(λ(ω)I −K∗D)−1[ν](x) dσ(x) ' (ν, φn)

λ(ω)− µn(x , ψn).

• SNR enhancement.

• Design of nano-particles: size, shape, coating (P. Millien).

• Gold nano-particles: selective accumulation in tumor cells;bio-compatibility; reduced toxicity.

• Detection: localized enhancement in radiation dose (strong scattering).

• Ablation: localized damage (high absorption).

• Functionalization: targeted drugs.

• Dilute suspension of nanoparticles: volume fraction f 1.

• Effective conductivity k∗: overall macroscopic material property of thecomposite material.

• (with H. Kang and K. Touibi)

k∗ =[I + f M(I − f

dM)−1] + o(f 2) ,

M: the polarization tensor associated with the (arbitrary shaped) scaledinclusion and the conductivity contrast k. The formula is uniform withrespect to the contrast.

• Maxwell Garnett (Clausius-Mossotti) formula (D disk or sphere); Fricke’sformula (D ellipsoid).

• k∗ blows up to infinity for some (negative) values of σ close to theeigenvalues of K∗D (in the smooth case).

Nano-particles for imaging• Dense suspension of nano-particles: target D with negative overall

parameters.

• Solution to the conductivity problem:

u = U + SD(λI −K∗D)−1[∂U

∂ν].

• SD : single layer potential.

• Spectral decomposition (D smooth): K∗D [ψn] = µnφn.

• ψn, φn: plasmonic eigenvectors.

• Spectral decomposition of the solution:

u − U =∑n

λ− µnSD [ψn]

• Far-field behavior:

u − U =∑n

λ− µn(∇G

∫∂D

xψndσ(x) + ∂2G

∫∂D

x2ψndσ(x) + . . .).

• G : Green’s function.

Membrane imaging

• Admittivities of biological tissues vary with the frequency of the appliedcurrent.

• Interface phenomena (cell membrane): super-resolution in electricalimaging of biological tissues (with L. Giovangigli).

• Cell: homogeneous core covered by a thin membrane of contrasting

electric conductivities and permittivities.

• Core: σext + iωεext (conducting effect; transport of charges);• Membrane: σint + iωεint with σint/σext 1 (capacitance

effect; storage or charges or rotating molecular dipoles);• Low frequencies: induced polarization effect due to the

membrane.• High frequencies: induced polarization effect disappears.

Membrane imaging

• δ: thickness of the membrane.

• Effective thickness:

ξ = δ(σint + iωεint)/(σext + iωεext).

• Electrical model of the cell:

∆u = 0 inD ∪ Rd \ D,

∣∣∣∣+

− ∂u

∣∣∣∣−

= 0 on ∂D,

u |+ −u |−= ξ∂u

∂νon ∂D.

Membrane imaging

• U: applied field. Integral representation of the potential u:

u = U +DD [ψ]

• DD : double-layer potential.

• Integral equation:

ξ∂DD

∂ν[ψ] + ψ = −ξ ∂U

∂ν,

ν: the outward normal to ∂D.

Membrane imaging

• Polarization tensor of the cell membrane:

M(ω) :=ξ

(σext + iωεext)

∫∂D

ν(ξ∂DD

∂ν+ I )−1[ν].

• Far-field behavior :

u(x)− U(x) ∼ −M(ω)∇U(z) · ∇G(x − z),

• G: Green’s function.

Membrane imaging

• Effective admittivity of a dilute suspension of cells:

σ∗(ω) = (σext + iωεext)[I + f M(ω)] + o(f ).

• Disk-shaped cells (D = |x | = r0):

DD [e inθ](x) =

)|n|e inθ if |x | = r < r0,

)|n|e inθ if |x | = r > r0.

• Maxwell-Wagner-Fricke’s formula:

M(ω) =δ

(σint + (δ/2r0)σext) + iω(εint + (δ/2r0)εext)I2.

Membrane imaging

• Dependence of the induced polarization on the frequency:

<eM ∝ ωτ 2

1 + ω2τ 2, =mM ∝ 1

1 + ω2τ 2,

• τ (Debye relaxation): the polarization does not occur instantaneously.

• <eM attains its maximum at ω = 1/τ .

• τ carries information on the microscopic parameters.

Concluding remarks

• Super-resolution in one single wave imaging:

• Differential imaging;• Spectroscopic imaging: target’s admittivity changes as a

function of the frequency.

• SNR enhancement: use of high-order polarization tensors (weakly electricfish); use of Plasmonic nano-particles induced resonances.

• Spectral induced polarization effects (weakly electric fish, cellmembranes).

• Plasmonic resonance of nano-particles.

• Effective medium theory : use of plasmonic resonances (the effectiveparameters blow up and the wavelength becomes much shorter).

• Physics-based classification.

Multi-wave imaging

• Ultrasound-modulated optical tomography (with E. Bossy, J. Garnier, L.Nguyen, and L. Seppecher).

• Thermo-acoustic tomography (with J. Garnier, W. Jing, and L. Nguyen).

Near infrared optical tomography

• Near infrared optical tomography: wavelengths 700− 1000nm,

• Differentiate between soft tissues: different absorption at the wavelengths.

• Absorption: dominated by oxy-hemoglobin, deoxy-hemoglobin, and water.

• Non-invasive (reasonable doses repeatedly employed), inexpensive.

Absorption spectrum.

Near infrared optical tomography

• µ′s : reduced scattering coefficient; µa: absorption coefficient;µa µ′s .

• Diffusion: −∆Φ + aΦ = 0 in Ω,

l∂νΦ + Φ = g on ∂Ω,

a(x) = 3µ′sµa(x), l : extrapolation length, g : the lightillumination on the boundary.

• Reconstruct a from boundary measurements of Φ.

• High contrast + low resolution.

Low resolution of optical tomography

NIR image of a breast tumor.

• Resolution enhancement: perturb the NIR light propagationby acoustic pulses inside the body and record the variation.

Ultrasound-modulated tomography

NIR light source

Light detectors

Focused acoustic beam

Acoustic source

Spherical acousticpulsesΩ y

Contrasted inclusion

• Record the variations of the light intensity on the boundarydue to the propagation of the acoustic pulses.

• Ω: acoustically homogeneous.

• Displacement field: spherical acoustic pulse generated at y .

• P : Ω −→ Ω: the displacement. u = P−1 − Id : smallcompared to |Ω|.

• Typical form of u:

uηy ,r (x) = −η r0rw

(|x − y | − r

)x − y

|x − y |, ∀x ∈ Rd .

• w : shape of the pulse; supp(w) ⊂ [−1, 1] and ‖w‖∞ = 1. η:thickness of the wavefront, y : source point; r : radius.

• Thin spherical shell growing at a constant speed.

• Pulse propagation: a→ au(x) = a(x + u(x)). Fluence Φu:−∆Φu + auΦu = 0 in Ω,

l∂nΦu + Φu = g on ∂Ω,

• au(x) = a(x + u(x)).

• Cross-correlation formula:

∫∂Ω

(∂νΦΦu − ∂νΦuΦ) =

(au − a)ΦΦu

u−Phi

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pulse radius

Φu − Φ (left); Mu (right).

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• u depends on the center y , the radius r and the wavefrontthickness η.

• Family of measurement functions:

Mη(y , r) =1

(auηy,r − a)ΦΦuηy,r

• Small η:

Mη(y , r) ≈ 1

∫Ω∇a.uηy ,rΦ2.

• Extract the information in Mη (asymptotically in η).

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True absorbtion (left) and measurements Mu (right)for 64 pulses centered on the unit circle.

• Asymptotic behavior:

limη→0

Mη(y , r) = −crd−2

∫Sd−1

(Φ2∇a)(y+rξ).ξdσ(ξ) =: M(y , r)

c > 0: depends on the shape of u and on d . Expansionuniform in (y , r); Error = O(η).

• M: ideal measurement function.

• Reconstruct a from M.

• Spherical means Radon transform:

R[f ](y , r) =

∫Sd−1

f (y + rξ)dσ(ξ) y ∈ S , r > 0,

• Derivative of R:

∂r (R[f ])(y , r) =

∫Sd−1

∇f (y + rξ) · ξdσ(ξ).

• Φ2∇a = ∇ψ: relate M to ∂rR[ψ] and then find ψ and Φ2∇afrom the measurements.

• Helmholtz decomposition of Φ2∇a:

Φ2∇a = ∇ψ +∇× A.

• Measurement interpretation:∫Sd−1

(Φ2∇a)(y + rξ).ξdσ(ξ) =

∫Sd−1

∇ψ(y + rξ).ξdσ(ξ).

• Reconstruction formula for ψ:

ψ = −1

cR−1

[∫ r

M(y , ρ)

ρd−2dρ

](up to an additive constant).

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True absorbtion a; Mu; R[ψ]; ψ.

• Reconstruct a knowing only ψ in the Helmholtzdecomposition:

Φ2∇a = ∇ψ +∇× A ?

• Divergence of the Helmholtz decomposition:

∇ · (Φ2∇a) = ∆ψ.

• Assume a = a0 (a known constant on Ω\Ω′):

(E2) :

∇ · (Φ2∇a) = ∆ψ in Ω′,

a = a0 on ∂Ω′.

• Φ: unknown in Ω.

Coupled elliptic system:

(E ) :

(E1) :

−∆Φ + aΦ = 0 in Ω

l∂nΦ + Φ = g on ∂Ω

(E2) :

∇ · (Φ2∇a) = ∆ψ in Ω′

a = a0 on ∂Ω′

a = a0 in Ω\Ω′

ψ, l > 0, g , and a0 > 0: known.

• Fixed point argument.

• Landweber scheme:• F [a] := ∇ · (Φ2[a]∇a);• Minimization problem: min ‖F [a]−∆ψ‖;• Landweber sequence:

a(n+1) = P(a(n))− µDF [P(a(n))]∗(F [P(a(n))]−∆ψ),

• µ > 0: relaxation parameter; P: projection.

• Convergence results.

• Minimal regularity assumption on a.

• Lipschitz stability results.

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True a, and reconstructions after 2 iterationswith 16, 32, 64 and 128 acoustic centers.

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Reconstruction of a from noisy measurements : true a;noise level: 0%, 5%, and 10%.

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Reconstruction of the Shepp-Logan phantom for 128 acoustic pulses.

Quantitative thermo-acoustic imaging

• Model: (∆ + k2 + ikq)u = 0 in Ω,

ν · ∇u − iku = g on ∂Ω.

• Reconstruct q from q|u|2 in Ω (thermal energymeasurements).

• The set (gj)d+1j=1 ⊂ L2(∂Ω): proper set of measurements (d :

space dimension) iff:

(i) |u1| > 0 in Ω.(ii) The matrix [uj ,∇Tuj ]1≤j≤d+1 is invertible for all x ∈ Ω.

• Ej := qu1uj : can be evaluated from the thermal energymeasurements.

• αj := Ej/E1, j = 2, . . . , d + 1.

• A = (∂lαj+1)j ,l=1,...,d : invertible (proper set ofmeasurements); a = A−1[(∇TAT )T ].

• Exact reconstruction formula:

q(x) =−<e(a) · =m(a) +∇ · =m(a)

• Exact formula: derivatives of the data (up to the third order).

• Noise regularization model (convolution with a smoothingkernel).

• Good initial guess.

• Resolution enhancement: optimal control approach.

Final concluding remarks

• One single wave imaging:

• Differential imaging .• High SNR: high sensitivity.• Spectral polarization and membrane effects: high specificity.• Plasmonic resonant nano-particles (high SNR, high effective

conductivity → high sensitivity, near-field imaging → highsensitivity + high specificity.

• Physics-based classification.

• Multi-wave imaging:

• Differential imaging.• Combination of two ways in one system: High sensitivity +

high specificity.• Exact reconstruction formula: good initial guess.

Sensitivity and specificity enhancement in medical imagingplc/ammari2012.pdfphoto-acoustic imaging...

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